In addition to knowing the value of a function f(x) at a
particular x, we often want to know how fast the function is changing with x. For a
straight line (linear function), this is simply the slope of the line.

In the case of a curved line, the slope is different at every point. In other words,
while the slope of a straight line is a constant number, the slope of a curved line is a
function of x. It is our business here to learn how to find and use this function, called
the derivative of f(x) and denoted f (x) (pronounced "f prime of
x").

There are a tremendous variety of curved lines. To precisely discuss them, well
have to build up a certain amount of mathematical machinery. In particular, we are going
to assume you have read the material on limits and worked a few examples.

The first idea to be considered is continuity. We say a function is continuous
if it has no sudden jumps or "missing" points. The graph of the function is
"connected" everywhere.

It is helpful to look at examples of discontinuities so we can
clearly see what continuity is not. The simplest possible discontinuous function is one
with a single step discontinuity:

This function is zero for x values less than zero and 1 for all x
greater than or equal to zero. It has a sudden step discontinuity at x = 0. Note that in
the graph below, the point (0, 0) is an open circle, indicating that that single point has
been left out of the function. The point (0, 1) on the other hand is a filled-in circle
and is included in the graph of f(x). This insures that the graph of the function conforms
exactly with the above definition.

Figure 1 - The Unit Step Function (step at 0)

Rigorously, the limit of this function as x approaches 0 from
below is different from the limit as x approaches 0 from above. Inspection of
the graph makes this clear. This function is clearly discontinuous at x = 0. Although f(x)
is continuous everywhere else, the single discontinuity makes the function discontinuous
(with respect to its domain, all real numbers).

Since this concept is complicated, lets go over the above
two expressions in detail.

The first statement holds that the limit of the function as x approaches x0
from the left must be the same as the limit as x approaches x0 from the right.
If this is true, then both limits are the same and we can refer to them collectively as
just the limit of f(x) as x approaches x0. In other words, if the left-handed
and right-handed limits are the same at a particular point, then there is a limit at that
point.

The second statement says that, if the limit exists, then it must be equal to the
function value at this point.

If we apply this idea to the step function above, we see that

This function fails the first condition (that the right and left
limits are the same), so it fails to be continuous at 0, and so is not a continuous
function.

There is another way a function can be discontinuous. Lets
look at a slightly different example:

This function is zero everywhere but x = 0, where it takes on the
value 1. This type of discontinuity is called a jump. The graph of this function looks
like this:

Figure 2 - The Unit Jump Function (jump at 0)

Notice that we have an open circle at the point (0, 0) to
indicate that the point has been excluded from the graph and a closed circle at (0, 1) to
show that this point as been included. If we perform the same analysis as above, we find
that

This function passes the first test of continuity since the right
and left-hand limits agree. However, if we compare the function value to the limit

the two values do not agree. This function is therefore not
continuous at this point and so is not continuous.

Now that we have considered a couple of counterexamples, we are
in a better position to look at a continuous function and consider how it is different.
Our example here will be

Figure 3 - Typical Continuous Function

For now, we will just assert that this function is continuous
everywhere and prove it in the course of the argument which follows. The important
background fact to remember here is that every real number has a real number square.
Since that is true, we really don't care which number we choose, so we'll go with a
generic real number. If the reader feels this is unclear, any real number can be used as
the example point.

Consider an arbitrary real number x0. Calculate the limit of f(x) as x
approaches x0 from the left and the right.

From elementary algebra, we know that x02
is defined whenever x0 is defined. In fact, we can calculate it explicitly for
rational x0 and approximate it to any (finite) degree of precision if x0
is irrational. If we also note that

Then we have the function value equal to the limit at the
arbitrary point x0. Since this point is arbitrary and the function value (and
limit) is defined, the function passes both tests everywhere and so is continuous
everywhere.

As we did above with continuity, it is instructive to consider a
function that is not differentiable so we can contrast it with functions that are
differentiable.

Consider the absolute value function

The graph of this function looks like this:

Figure 4 - The Absolute Value Function

The interesting point is at zero  this function has a
sharp, sudden kink to it. By inspection, it is pretty clear that the slope of this
function to the left of zero is m = -1, but at zero or above the slope is m = +1:

Figure 5 - Derivative of Absolute Value Function

The slope is not continuous at zero  we say that this
function is not differentiable at zero. When we look at the definition of the
derivative below, it will be easy to see that the left and right hand limits of the
derivative function must match at a point in order for the derivative to exist at that
point. Moreover, the value of the derivative must match what we expect from the limits to
get a properly defined derivative at that point. In other words, the derivative must be a
continuous function.

From elementary algebra, we have the familiar formula for the
slope of a line given two points on the line (x1, y1) and (x2,
y2):

Definition of Slope
(two-point form)

Were we to use functional notation with this formula, in other
words use the substitution

to get

Definition of Slope (functional form)

then it becomes a bit clearer that we are actually dividing an
increment in the function value by an increment in the variable. As it stands now, this
number is the same everywhere for a straight line because the ratio of the rise to the run
is always constant. For a more general function, the value m above depends in general on
both x1 and x2; different choices of these two numbers will give
different values of m.

What we would like to do is find the slope of a function at a point x0 as a
function of that point. In that case, we will redefine our points a bit, to allow us to
better calculate this number.

Put

so that the slope now looks like

If we let x approach x0, in other words, take the
limit, we have the definition of the derivative:

We can do this a different way and get the same results. This
time, make the following substitutions in the functional form of the slope definition:

Substituting these into the slope formula gives

In this last expression, x0 is (again) the point where
we wish to find the slope and h is now an increment (either positive or negative) that
defines how far away from x0 we move to get another point to calculate the
slope. You can think of h as a dial that we can turn to get various approximations of our
slope. The smaller the absolute value of h, the closer we get to our derivative. However,
since h appears alone in the denominator, we cannot just substitute zero  the
expression would then be undefined.

It should come as no surprise that we use the limiting process to "dial down"
the value of h. This then becomes the alternativedefinition of
the derivative:

Here we consider the function x2 (graphed in blue).
First, we take two points on the x-axis, x0 and some other point x. For each x
value find the function values f(x0) and f(x). This gives us two points on the
graph of x2: (x0, f(x0)) = (x0, x02)
and (x, f(x)) = (x, x2). Draw a straight line through them (graphed in red).

Figure 6 - Construction of Derivative

Imagine how the line graphed in red changes as we move x a little
closer to x0. The slope becomes a bit less steep, and the y-intercept of the
line moves up a little bit. When x is moved all the way over so that x = x0 the
graph looks like this:

Figure 7 - Derivative as Tangent Line

Now the red straight line touches the graph of x2 at
only one point (x0) and has the same slope as the curve at this single point
only. We call this straight line the tangent line at x0. The slope
of this tangent line is the value of the derivative of x2 at x0.

Inspection of the preceding examples suggests (but does not completely prove) the
following formula:

We showed the calculations above for integer values of n and one
example of a rational value of n. The formula is good for general ns (any real n),
but the proof of the most general case is too involved for our purposes here. Use of this
formula considerably speeds calculation of derivatives for powers of x.

We will use this formula whenever possible since it is much
easier to use than the definition. The reader should be cautioned that the magic formula
does not work everywhere. In particular, the magic formula does not work for exponential
functions.

Exponential functions raise a positive number to the power of x, rather than x raised
to a number power:

Fortunately, we do have ways of dealing with exponential
functions elegantly and will cover them later.

When in doubt, start with either version of the definition. The definition is
guaranteed to be correct, but not guaranteed to produce manageable algebra.